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I want guidance in relation to geometrical aspects about the image I've added. The part where the man's got his right hand is called 'edge' or 'end' 'horizontal border' or 'border'? Similarly, the place where the man grips the sheet with his left hand is called 'end' or something else? enter image description here

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    I would call that the 'edge' of the sheet. The word 'border' applies more to art, web pages, country borders, ornamentation. The left hand is on the corner. – Sam May 12 '18 at 10:04
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The man's right hand is gripping an edge (the top or upper edge) of the sheet; his left hand is gripping a corner (the top left corner as it appears to him).

edge noun (OUTER POINT) ​ B1 [ C ] the outer or furthest point of something:

He put pink icing around the edge of the cake. They built the church on the edge of the village. A man was standing at the water's edge with a small boy. I caught (= hit) my leg on the edge of the table as I walked past.

https://dictionary.cambridge.org/dictionary/english/edge

corner noun [ C ] UK ​ /ˈkɔː.nər/ US ​ /ˈkɔːr.nɚ/ ​ A2 the point, area, or line that is formed by the meeting of two lines, surfaces, roads, etc.:

You go around corners too fast when you're driving! There's a postbox on the corner (= the place where the street crosses another). Click the icon in the bottom right-hand corner of the screen. I have a bruise where I hit my leg on the corner of the table. They live just around/round the corner (= very close although not in the same road) - so we see them all the time.

https://dictionary.cambridge.org/dictionary/english/corner

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